Correspondence of Fixed-Point Theorem in \(T_2, T_3\)-SPACE

Abstract

Fixed-point theory (FPT) has lot of applications not only in the field of mathematics but also in various other disciplines. Fixed Point Theorem presents that if T:X \(\to\) X is a contraction mapping on a complete metric space (X, d) then there exists a unique fixed point in X. FPT is also essential in game theory, in this case Brower Fixed Point has an application in game theory specifically in non-cooperative games and existence of Equilibrium. In particular, a game is a set of actions done by the participants defined by a set of rules. This is commonly described using mathematical concepts, which offers a concrete model to describe a variety of situations. On the other hand, the separation axioms Ti, i = 0,1,2,3,4 are vital properties that describes the topological spaces T0,T1,T2,T3 and T4. It is noted that a T3 - space is a generalized version of T2- space and since various results on application of fixed point theory in game theory on an arbitrary locally convex T2 - space has been established, in this study we sort to extend this concept to the general T3 - space. The utilization of a symmetric property of Hausdorff space established that if two continuous commutative mappings are defined on a T3 - space, then the two maps achieves unique fixed points.